I am going off the beaten path here.
your heater gets a signal from the thermostat when it is calling for heat.
your gas valve opens and the flow starts, a thermocouple is needed with a pilot, or you have an pilot-less and start the fire each time.
once the heat exchanger/combustion chamber starts to get hot, the fan starts.
your arduino should be able to tell when the heat is on. a rise in the heat exchange. temperature sensors galore ! for the Arduino.
you can monitor that hte fan is on. lots of ways to do that as well.
log your run time. after X hours, go look at the tank.
each time you fill, note how many hours run time you had.
no need to actually see what the tank level is for most of the information.
as for getting a sense of full, after a few fills, you should have a good feel for calibrating the hours to usage rate.
my oil burner has a motor on the oil burner itself. I know that an hour dumps X gallons of oil.
the heat exchanger fan is not 1:1 with the burner motor. not sure if you can monitor the gas valve.
a simple 10 point LED fuel gauge can be mounted on your Arduino to show what it thinks is the tank level.
the real goal, might be to know once you are less than half full ?
and doing all this should be a simple and fun exercise.